Statistical causality, optional and predictable projections*.

We consider the statistical concept of causality in continuous time within filtered probability spaces based on Granger's definition of causality. The given causality concept is connected to the optional and predictable processes important in stochastic integration. More precisely, we establish that the preservation of predictability with respect to larger filtrations is implied by the considered notion of (self-)causality. We also consider the connections between the given causality concept and the optional and predictable projections of a stochastic process, which play an important role in the general theory of stochastic processes, semimartingale theory, and stochastic calculus. Some results show that the (self-)causality implies indistinguishability of the optional (or predictable) projections with respect to considered filtrations from those with respect to larger filtrations. [ABSTRACT FROM AUTHOR]